Full-revivals in 2-D Quantum Walks

TL;DR

This paper demonstrates that 2-D quantum walks can exhibit full state revivals and stationary solutions due to localization effects, highlighting quantum interference phenomena absent in classical walks.

Contribution

It shows that 2-D Grover walks can have full state revivals and stationary states, establishing the existence of short cycles and the role of interference in quantum recurrence.

Findings

Full revivals with a period of two steps in 2-D Grover walks.

Longer cycles are impossible for four-state quantum walks.

Localization enables stationary states and revivals due to quantum interference.

Abstract

Recurrence of a random walk is described by the Polya number. For quantum walks, recurrence is understood as the return of the walker to the origin, rather than the full-revival of its quantum state. Localization for two dimensional quantum walks is known to exist in the sense of non-vanishing probability distribution in the asymptotic limit. We show on the example of the 2-D Grover walk that one can exploit the effect of localization to construct stationary solutions. Moreover, we find full-revivals of a quantum state with a period of two steps. We prove that there cannot be longer cycles for a four-state quantum walk. Stationary states and revivals result from interference which has no counterpart in classical random walks.